## In Bayesian statistics, data is considered nonrandom…

**A** rather weird question popped up on X validated, namely why does Bayesian analysis rely on a sampling distribution if the data is nonrandom. While a given sample is is indeed a deterministic object and hence *nonrandom* from this perspective!, I replied that on the opposite Bayesian analysis was setting the observed data as the realisation of a random variable in order to condition upon this realisation to construct a posterior distribution on the parameter. Which is quite different from calling it *nonrandom*! But, presumably putting too much meaning and spending too much time on this query, I remain somewhat bemused by what line of thought led to this question…

July 12, 2021 at 1:38 pm

I think that question confuses ontological/epistemic assumptions, and hence why the question seems bizarre. If the former, Bayesians should be agnostic about, or flat out reject, the assumption that data are actually generated from some distribution (ie random), since with it we cannot justify its usage for non-random events. Savage talks about this (eg whether Chicago is north of NY), Lindley talks about this (eg court cases and guilt), and Good talks about this (in “Kinds of probability” 1959, Good argues that, if the world is fully deterministic, then Bayes would be the only valid approach). In fact, it seems like this has historically been the motivation of much of subjective Bayes. Simply, being agnostic about it expands the scope of Bayesian paradigm beyond that of frequentism. If the latter, the fact that we use a parametric assumption on the data (as with your answer) is not our assumption on the data itself but rather reflects our epistemic uncertainty on the (un/observed) data, which is not the same thing as a frequentist saying the data are random (in fact, this tension within frequentism is what Jim said was absurd about the procedural frequentist).

Of course, I feel like you know all of this, which is why I’m confused why you answered (paraphrasing) “Bayesians and frequentists treat data the same way,” which would be false from a subjective Bayesian perspective. Or perhaps I’m misunderstanding your point in that answer.